Max Planck Institute for Physics, Munich, Germany · T. Ahmed, W. Bernreuther, A. H. Ajjath, P. K....

Some issues in the computation of 2-loop amplitudes qq→ ZH

Long Chen

Max Planck Institute for Physics, Munich, Germany

Seminar, Berlin, January 9

T. Ahmed, W. Bernreuther, A. H. Ajjath, P. K. Dhani, P. Mukherjee, V. Ravindran,based on JHEP01(2020)030 [arXiv: 1910.06347] and work in progress

h(125)→ γγ

The discovery of the h(125) at the LHC (2012)

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Long Chen (MPI) bb→ ZH HU Berlin 1 / 28

h(125)→ bb observed through VH(bb)

h→ bb finally observed recently at the LHC in the VH-events!

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Much theoretical work done already ...

Much work done on the ZH production at LHC: P + P→ Z(l l′) + H(bb)

qq→ ZH: Starting from O(αe α0

s )

I The Higgs-bremsstrahlung (Drell-Yan) part up to NNLO in massless QCD [Brein,

Harlander,Wiesemann,Zirke, 2012; Ferrera,Grazzini,Tramontano, 2015/2018; Campbell,Ellis,Williams, 2016]I The top-loop induced NNLO (non-Drell-Yan type) QCD corrections in the heavy-top

limit [Brein,Djouadi,Harlander, 2004; Brein,Harlander,Wiesemann,Zirke, 2012]I N3LO corrections in massless QCD [Ahmed,Mahakhud,Rana,Ravindran, 2014; Li,von Manteuffel, Schabinger,Zhu,

2014; Catani,Cieri,Florian,Ferrera,Grazzini, 2014; Kumar,Mandal,Ravindran, 2015]I NLO electroweak corrections [Ciccolini,Dittmaier,Kramer, 2003; Denner,Dittmaier,Kallweit,Mueck, 2011]I · · · · · ·

gg→ ZH: Starting from O(αe αs)

I The exact LO (with full mt dependence) [Kniehl, 1990]I NLO QCD corrections to the Higgs-bremsstrahlung (Drell-Yan) part in the heavy-top

limit [Brein, Harlander,Wiesemann,Zirke, 2012; Altenkamp, Dittmaier, Harlander, Rzehak, Zirke, 2013]I · · · · · ·


Focus of the talk

Here we focus on a part of the b-quark-induced ZH process that involves anon-vanishing Yukawa coupling λb (i.e. (B) and (C)) but with mb = 0.

Aims of the work:

Computing the 2-loop (massless) QCD corrections in analytic form

Addressing a subtlety appearing in the conventional FF decomposition of amplitudesinvolving axial currents regularised in D dimensions (with a non-anticommuting γ5)

Verifying the unitarity of a particular regularisation prescription implied by projectorsprescribed recently [Chen, 2019]]

The “same” loop amplitudes built up from just vector FFs of properly grouped classes ofdiagrams (bypassing completely the need of explicitly manipulating γ5)


Kinematics and the γ5 prescription

We consider

b(p1) + b(p2)→ Z(q1) + H(q2)

in nf = 5 massless QCD.

The Mandelstam variables:

s ≡ (p1 + p2)2 , t ≡ (p1 − q1)

2 and u ≡ (p2 − q1)2

satisfying s + t + u = q2

1+ q2

2= m2

z + m2

h.

Regarding the definition of γ5 in dimensional regularisation (HV/BM and Larin):

γ5 =i

4!εµνρσγµγνγργσ

γµγ5 →1

2

(γµγ5 − γ5γµ

)=

i6

εµνρσγνγργσ .


Form Factor Decomposition: the vector part

By Lorentz covariance, the amplitude bbZH can be expressed as alinear combination of a finite basis of Lorentz structures at anyfinite order.

For the vector part :

v(p2) Γµvec u(p1) = F1,vec v(p2) u(p1) pµ

1 + F2,vec v(p2) u(p1) pµ2

+ F3,vec v(p2) u(p1) qµ1 + F4,vec v(p2) γµ/q1

u(p1) ,

under the following constraints:

Even power γ-matrices (one Yukawa vertex on the massless fermion line);

Parity even for the vector coupling;

Equations of motion for the on-shell massless spinors: /p1u(p1) = 0 , /p2

v(p2) = 0.


Form Factor Decomposition: Gram matrix and Projectors

For a scattering amplitude M at a fixed perturbative order,

M =N

∑n=1

Fn Tn ,

compute the Gram matrix G

Gij = 〈T†i , Tj〉 ,

whose inverse gives us the formula for the projector

Pn = G−1

nj T†j

to project out the Lorentz-invariant coefficient (form-factor) Fn.


Projectors for vector form factors

The projectors for vector form factors are still compact enough to be documentedexplicitly:

Pµ1,vec = u(p1)

{(−2 + D)t2pµ

1+(

2(−3 + D)m2

zs− (−2 + D)tu)

pµ2+ (2−D)stqµ

1

+ st /q1γµ}

v(p2)1

Kvec,

Pµ2,vec = u(p1)

{(2(−3 + D)m2

zs− (−2 + D)tu)

pµ1+ (−2 + D)u2pµ

2+ (4−D)suqµ

1

− su /q1γµ}

v(p2)1

Kvec,

Pµ3,vec = u(p1)

{− (−2 + D)stpµ

1− (−4 + D)supµ

2− (2−D)s2qµ

1− s2 /q1γµ

}v(p2)

1

Kvec,

Pµ4,vec = u(p1)

{stpµ

1− supµ

2− s2qµ

1+ s2 /q1γµ

}v(p2)

1

Kvec,

where

Kvec = 2(−3 + D)s(m2

zs− tu) .


Form Factor Decomposition: issues with the axial part

A set of Lorentz structure basis for the axial part of bb→ ZH linearly complete in 4dimensions, but not in D dimensions (unless a fully anticommutating γ5 is used):

{v(p2) γ5 u(p1) pµ

1, v(p2) γ5 u(p1) pµ

2, v(p2) γ5 u(p1) qµ

1, v(p2) γµγ5 /q1

u(p1)}

.

WhetherMaxi lives in a space linearly spanned by these 4 structures in D dimensions:

I Enlarge this list by appending the (tree-level)Maxi;I and then compute the Gram matrix of this enlarged

list of 5 elements;I one finds out that the matrix rank is increased to 5

rather than staying at 4!


Form Factor Decomposition: issues with the axial part

A set of Lorentz structure basis for the axial part of bb→ ZH linearly complete in 4dimensions, but not in D dimensions (unless a fully anticommutating γ5 is used):

{v(p2) γ5 u(p1) pµ

1, v(p2) γ5 u(p1) pµ

2, v(p2) γ5 u(p1) qµ

1, v(p2) γµγ5 /q1

u(p1)}

.

Therefore, if a non-anticommuting γ5 is used in the dimensional regularisation, we mayface the following issues regarding the axial FF-decomposition :

1 It is not easy to construct the full D-dimensional linearly complete basis for theFF-decomposition of v(p2) Γµ

axi u(p1) at loop orders.

2 Even with just an incomplete basis, keeping the full D-dependence could lead toexpressions with too complicated D-dependence to use.

3 Setting D=4 could simplify the expressions a bit, which, however, is not known tobe legitimate in general (for FF-decomposition).


Projectors for axial form factors

The projectors for 4 axial FFs (corresponding to the basis of 4 structures) read as:

Pµ1,axi = −u(p1 )εγγγγ v(p2 )

{(−1 + D)2 (48− 14D + D2 )(m2

z − u)2 pµ1

+((176 + 22D− 69D2 + 16D3 −D4 )m4

z

+ 2(−148− 54D + 73D2 − 16D3 + D4 )m2

z s

+ (176 + 22D− 69D2 + 16D3 −D4 )tu

− (176 + 22D− 69D2 + 16D3 −D4 )m2

z (t + u))

pµ2

− (−1 + D)2 (48− 14D + D2 )s(m2

z − u)qµ1

}12

(8− 9D + D2 )s2Kaxi

+ u(p1 ) /q1 εγγγµ v(p2 )

{(−56 + 71D− 16D2 + D3 )s(m2

z − u)

}12

(8− 9D + D2 )s2Kaxi,

Pµ2,axi = u(p1 )εγγγγ v(p2 )

{((−176− 22D + 69D2 − 16D3 + D4 )m4

z

−−2(−148− 54D + 73D2 − 16D3 + D4 )m2

z s

+ (−176− 22D + 69D2 − 16D3 + D4 )tu

− (−176− 22D + 69D2 − 16D3 + D4 )m2

z (t + u))

pµ1

− (m2

z − t)2 (−144− 62D + 77D2 − 16D3 + D4 )pµ2

− (m2

z − t)(120 + 86D− 77D2 + 16D3 −D4 )sqµ1

}12

(8− 9D + D2 )s2Kaxi

+ u(p1 ) /q1 εγγγµ v(p2 )

{(−1 + D)(m2

z − t)

}12

sKaxi,


{− (−1 + D)2 (48− 14D + D2 )(m2

z − u)pµ1

− (−120− 86D + 77D2 − 16D3 + D4 )(m2

z − t)pµ2

+ (−1 + D)2 (48− 14D + D2 )sqµ1

}12

(8− 9D + D2 )sKaxi

− u(p1 ) /q1 εγγγµ v(p2 )

{(−56 + 71D− 16D2 + D3 )s

}12

(8− 9D + D2 )sKaxi,


{− (−7 + D)(m2

z − u)pµ1

− (−1 + D)(m2

z − t)pµ2 + (−7 + D)sqµ

1

}12

sKaxi

− u(p1 ) /q1 εγγγµ v(p2 )(8− 9D + D2 )3

Kaxi,

Kaxi = (−888 + 416D + 560D2 − 515D3 + 159D4 − 21D5 + D6 )

(m4

z + tu−m2

z (s + t + u)) ,

εγγγµ ≡ − i6

ενρσµ γν γρ γσ .


An acrobatic form factor decomposition for the axial part

Due to the aforementioned theoretical issues, we define our “axial form factors” as

F1,axi ≡ P

[4], µ1,axi v(p2) Γν

axi u(p1) gµν

F2,axi ≡ P


axi u(p1) gµν

F3,axi ≡ P


axi u(p1) gµν

F4,axi ≡ P


axi u(p1) gµν

where the [4] in the superscript denotes the setting D = 4 in the original Pµi,axi.

Subsequently, we build up an intermediate axial amplitude Mµaxi defined as

Mµaxi ≡ F

1,axi v(p2) γ5 u(p1) pµ1+ F

2,axi v(p2) γ5 u(p1) pµ2

+ F3,axi v(p2) γ5 u(p1) qµ

1+ F

4,axi v(p2) γµ γ5 /q1u(p1) .

which is not algebraically identical to the original Feynman amplitude.


Projectors for linearly polarised amplitudes

Projecting D-dimensional amplitudes directly onto a linear polarisation basis [LC,

19 (arXiv: 1904.00705)] :

Momentum basis representations of elementary linear polarisation vectors:

εµX = cX

1pµ

1+ cX

2pµ

2+ cX

3pµ

3,

εµT = cT

1pµ

1+ cT

2pµ

2+ cT

3pµ

3,

εµY =

{NY ενρσµp1νp2ρp3σ if N ≤ 4

cY1

pµ1+ cY

2pµ

2+ cY

3pµ

3+ cY

4pµ

4if N > 5



For bbZH, we need only

u(p1)Ni v(p2) εµj , for i = s, p and j = T, Y, L

where all open Lorentz indices are D-dimensional by definition andall pairs of ενρσµ should be contracted first (in one definite ordering).

Upon pulling out the overall normalization factors, all projectors soconstructed have only polynomial dependence in kinematics, and itis always gµν used in index contraction.

Resulting (bare) amplitudes are different from those defined inCDR, HV, FDH, DRED, · · · , albeit unitarity is still preserved.

The usual helicity amplitudes can be constructed optionally, ascircular polarisation states from the linear ones, e.g.

εµ±(p1) =

1√2

(ε

µX ± iεµ

Y)

.



Up to overall normalization factors, our “linear polarisation” projectors for bbZH read:

Pµ1= u(p1) v(p2)

(−(2m4

z + u(t + u)−m2

z(2s + t + 3u))

pµ1

+(2m4

z + t(t + u)−m2

z(2s + 3t + u))

pµ2+ s(t− u) qµ

1

),

Pµ2= u(p1) v(p2)

(− εµνρσp1 νp2 ρq1 σ

),

Pµ3= u(p1) v(p2)

((2m2

z − t− u) qµ1− 2m2

z(pµ1+ pµ

2))

,

Pµ4= u(p1)εγγγγv(p2)

(−(2m4

z + u(t + u)−m2

z(2s + t + 3u))

pµ1

+(2m4

z + t(t + u)−m2

z(2s + 3t + u))

pµ2+ s(t− u) qµ

1

),

Pµ5= u(p1)

i8

((−2m2

z + t + u)(

/p2γµ + γµ/p1

)+ 2s

(γµ/q1

− /q1γµ

)+ 2(u− t)

(p1µ + p2µ

))v(p2) ,

Pµ6= u(p1)εγγγγv(p2)

((2m2

z − t− u) qµ1− 2m2

z(pµ1+ pµ

2))

.


The tool chain employed for computing (projected) amplitudes

Unreduced (projected) amplitudes: QGRAF [P. Nogueir, 1993] + FORM [J.

Vermaseren, 2000]

IBP-tables: Kira [P. Maierhofer, J. Usovitsch, P. Uwer; 17/18]

I 8 one-loop masters known to O(ε2)

I 134 two-loop masters known to O(ε0).

All these masters are available in HepForge in computer readableformat [J. Henn, K. Melnikov, V. A. Smirnov, 2014; T. Gehrmann, A. von Manteuffel, L. Tancre, 2015]

Simplifying rational coefficients of masters: mathematica +fermat [R. Lewis; 2009]


Samples of loop diagrams

1-loop:

2-loop flavor-nonsinglet (non-anomalous):

2-loop flavor-singlet (anomalous):


UV renormalisation

Vector part: the MS scheme.I The QCD coupling:

asSε = as(µ2

R)Zas (µ2

R)

(µ2

µ2

R

)−ε

I The Yukawa coupling:

λbSε = λb(µ2

R)Zλ(µ2

R)

(µ2

µ2

R

)−ε

Axial part: additional axial-current ren. Jns(s)µ,A (x) = Zns(s)

5,A ZnsA Jns(s)

µ,A (x) [M. Chanowitz,

M. Furman, I. Hinchliffe, 1979; T. Trueman, 1979].I Flavor-Nonsinglet (non-anomalous) [S. Larin, J. Vermaseren, 1991]:

ZnsA = 1 + a2

s (µ2

R)1

ε

(22

3

CFCA −4

3

CFnf

),

Zns5,A = 1 + as(µ

2

R) (−4CF) + a2

s (µ2

R)

(22C2

F −107

9

CFCA +2

9

CFnf

).

I Flavor-Singlet (anomalous) [S. Larin, 1993]:

ZsA = 1 + a2

s (µ2

R)3

εCF ,

Zs5,A = 1 + a2

s (µ2

R)3

2

CF .


UV renormalisation

Vector part: the MS scheme.I The QCD coupling:

asSε = as(µ2

R)Zas (µ2

R)

(µ2

µ2

R

)−ε

I The Yukawa coupling:

λbSε = λb(µ2

R)Zλ(µ2

R)

(µ2

µ2

R

)−ε

Axial part: additional axial-current ren. Jns(s)µ,A (x) = Zns(s)

5,A ZnsA Jns(s)

µ,A (x) [M. Chanowitz,

M. Furman, I. Hinchliffe, 1979; T. Trueman, 1979].

UV renormalised amplitudes:

M[j] =M[j]vec(as(µ

2

R)) +M[j]axi(as(µ

2

R))

=M[j]vec(as(µ

2

R)) +M[j],nsaxi (as(µ

2

R)) +M[j],saxi (as(µ

2

R))

= M[j]vec(as, µ2) + Zns

5(as(µ

2

R))ZnsA (as(µ

2

R))M[j],nsaxi (as, µ2)

+ Zs5(as(µ

2

R))ZsA(as(µ

2

R))M[j],saxi (as, µ2) ,

where [j] runs over all six polarisation configurations.


IR factorisation formulae

The IR pole structures in the UV renormalised bbZH amplitudes can be exhibitedthrough a factorisation formula in terms of “universal” I(i)(ε) [S. Catani, 1998]

M[j],(1) = 2I(1)(ε)M[j],(0) +M[j],(1)fin ,

M[j],(2) = 4I(2)(ε)M[j],(0) + 2I(1)(ε)M[j],(1) +M[j],(2)fin .

M[j],(1)fin andM[j],(2)

fin (in 4 dimensions) are defined as the finite remainders.

The explicit expressions of the I(i)(ε) (needed for bbZH) are given (in CDR) by [S. Catani,

1998; T. Becher, M. Neubert, 2009]

I(1)(ε) = −CFeεγE

Γ(1− ε)

(1

ε2+

3

2ε

)(−

µ2

Rs

)ε

,

I(2)(ε) = −1

2

I(1)(ε)(

I(1)(ε) +1

εβ0

)+

e−εγE Γ(1− 2ε)

Γ(1− ε)

(2

εβ0 + K

)I(1)(2ε) + 2H(2)(ε)

IR poles contained in our renormalised amplitudes match with those predicted by these formulae.Long Chen (MPI) bb→ ZH HU Berlin 20 / 28

The same RS-independent finite remainders

The linear transformation connecting Fi,vec to Pµi v(p2) Γvec,µ u(p1) reads as

Pµi v(p2) Γvec,µ u(p1) = F1,vec

(Pµ

i v(p2) u(p1) p1,µ

)+ F2,vec

(Pµ

i v(p2) u(p1) p2,µ

)+ F3,vec

(Pµ

i v(p2) u(p1) q1,µ

)+ F4,vec

(Pµ

i v(p2) γµ/q1u(p1)

),

and similarly for the axial part,[Pµ

i v(p2) Γaxi,µ u(p1)]

fin = F1,axi,fin

[Pµ

i v(p2) γ5 u(p1) p1,µ

]+ F

2,axi,fin

[Pµ

i v(p2) γ5 u(p1) p2,µ

]+ F

3,axi,fin

[Pµ

i v(p2) γ5 u(p1) q1,µ

]+ F

4,axi,fin

[Pµ

i v(p2) γµγ5 /q1u(p1)

],

where i runs from 1 to 6 linear polarisation configurations.

Through the verified equality between the finite remainders computed followingdifferent approaches,

we confirm:projectors derived in “four” dimensions can be used also in calculations inD dimensions and lead to correct results (for physical observables),irrespective of whether the quantity projected out is a form factor or alinearly polarised amplitude.

(even though the resulting amplitudes may not be regularised strictly in the HV scheme.)Long Chen (MPI) bb→ ZH HU Berlin 21 / 28

Axial Form factors restored from Vector Form Factors

With mb = 0, all 2-loop diagrams with the Higgs (and Z) radiated from a closed fermionloop vanish, e.g.

due to odd number of Dirac γ matrices.

Consequently, all non-vanishing non-anomalous Feynman diagrams can be dividedinto the class-HZ and class-ZH


Axial Form factors restored from Vector Form Factors

Turning off completely the axial coupling of the Z boson,

Mvec = v(p2) ΓµZH u(p1) ε∗µ(q1) + v(p2) Γµ

HZ u(p1) ε∗µ(q1)

v(p2) ΓµZH u(p1) = F

1,ZH v(p2) u(p1) pµ1+ F

2,ZH v(p2) u(p1) pµ2

+ F3,ZH v(p2) u(p1) qµ

1+ F

4,ZH v(p2) γµ/q1u(p1) ,

v(p2) ΓµHZ u(p1) = F

1,HZ v(p2) u(p1) pµ1+ F

2,HZ v(p2) u(p1) pµ2

+ F3,HZ v(p2) u(p1) qµ

1+ F

4,HZ v(p2) γµ/q1u(p1) .

Axial (and vector) Form factors can be restored as:

Fi,vec = Fi,HZ + Fi,ZH ,

Fi,axi(ns) = Fi,HZ− Fi,ZH .


The Ward identity for bbZH: the non-anomalous part

Starting from the classical Lagrangian,

Lc = − 1

4Ga

µνGaµν + biγµDµb− JµZZµ − λbbbH ,

one obtains the Ward identity for the non-anomalous diagrams:

qµ1Mµ = −2gA,bλb〈H(q2)|b(0)iγ5b(0)H(0)|b(p1)b(p2)〉 .

whose RHS can be perturbatively expanded as


Checking the Ward identity for bbZH using Fi,ZH and Fi,HZ

qµ1Mµ = −2gA,bλb〈H(q2)|b(0)iγ5b(0)H(0)|b(p1)b(p2)〉 .

The LHS of the Ward identity can be composed in terms of “split” vector form factors:

q1 ·Mvec = v(p2) u(p1)

((F

1,HZ + F1,ZH

) m2

z − t2

+(

F2,HZ + F

2,ZH) m2

z − u2

+(

F3,HZ + F

3,ZH)

m2

z +(

F4,HZ + F

4,ZH)

m2

z

),

q1 ·Maxi = v(p2) γ5 u(p1)

((F

1,HZ − F1,ZH

) m2

z − t2

+(

F2,HZ − F

2,ZH) m2

z − u2

+(

F3,HZ − F

3,ZH)

m2

z +(

F4,HZ − F

4,ZH)

m2

z

).


The Ward identity for bbZH: the anomalous part

The Adler-Bell-Jackiw (ABJ) anomaly equation:(∂µJs

µ,A

)R= as

1

2

(GG)

R ,

where GG ≡ εµνρσGaµνGa

ρσ with the gluonic field strength tensor Gaµν.

At the level of matrix elements (regarding the anomalous part of bb→ ZH),

v(p2) ΓµABJ u(p1) q1,µ = gA,q

as

2

⟨H(q2)

∣∣[GG(0)]

R

∣∣q(p1)q(p2)⟩

,

(with the kinematics p1 + p2 − q2 = q1).

Diagrammatically, the RHS consists of


Numerical results

The numerical results of the total cross section of ZH production atLHC@13TeV (in unit pb):

Order s-channel EW σZHgg σZH

qq (top) (t + u)-channel

LO 5.897 10−1 -3.111 10−2 - - 2.989 10−4

NLO 7.756 10−1 - - - 2.934 10−4

NNLO 8.015 10−1 - 5.051 10−2 9.442 10−3 3.027 10−4

N3LOSV 8.013 10−1 - - - -

Setting: µR = µF = mh + mz, GF = 1.16637× 10−5 GeV−2, mb(µR = mb) = 4.18 GeV.

I The s-channel contributions (Higgs-bremsstrahlung) are obtained usingvh@nnlo [Brein,Harlander,Zirke, 2012].

I The σZHgg refers to the contribution coming from the gluon initiated sub-processes.

I The top quark loop contribution is denoted by σZHqq (top).

I The (t + u)-channel contribution at NLO is obtained usingMadgraph [Alwall,Frederix,Frixione,Hirschi,Maltoni,Mattelaer et al.,2014], and at NNLO is under the Soft-Virtualapproximation [Ravindran,2005].


Summary

2� Computed the 2-loop QCD corrections to bb→ ZH amplitude via directlyprojecting onto a linear polarisation basis, with the analytic results expressed interms of multiple polylogarithms;

2� Addressed an interesting subtlety appearing in the conventional form-factor(FF)decomposition of amplitudes involving axial currents regularised in D dimensions;

2� Revealed a relation between axial and vector FFs of the non-Drell-Yan bb→ ZH,which enables us to restore axial FFs from “split” vector ones;

2� Derived the Ward identities for bb→ ZH in the presence of λb, which are checkedusing the axial FFs restored from their vector counterparts;

2� Computed the SV cross section at NNLO, in order to make a quantitative analysisof the contribution from these non-Drell-type processes.


Summary

2� Computed the 2-loop QCD corrections to bb→ ZH amplitude via directlyprojecting onto a linear polarisation basis, with the analytic results expressed interms of multiple polylogarithms;

2� Addressed an interesting subtlety appearing in the conventional form-factor(FF)decomposition of amplitudes involving axial currents regularised in D dimensions;

2� Revealed a relation between axial and vector FFs of the non-Drell-Yan bb→ ZH,which enables us to restore axial FFs from “split” vector ones;

2� Derived the Ward identities for bb→ ZH in the presence of λb, which are checkedusing the axial FFs restored from their vector counterparts;

2� Computed the SV cross section at NNLO, in order to make a quantitative analysisof the contribution from these non-Drell-type processes.

T H A N K Y OU


Max Planck Institute for Physics, Munich, Germany · T. Ahmed, W. Bernreuther, A. H. Ajjath, P. K....

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Transcript of Max Planck Institute for Physics, Munich, Germany · T. Ahmed, W. Bernreuther, A. H. Ajjath, P. K....